The elastic properties of the materials used in many fields are often critical to the design, operation, utility, or safety of the uses of these materials. In the field of manufacturing, the elastic properties of manufactured materials and their components often must meet defined specifications which are essential to the utility and safety of the manufactured products. In the medical field, elastic properties of biological tissues is important for ultrasound imaging. In the field of construction, the elastic properties of construction materials and of foundation soils are important design criteria and critical safety considerations for engineered structures, roads, dams, excavations, and earthworks. In the field seismic surveying, the measurement of spatial variation and depth variation of near-surface elastic wave speed in the earth is important for seismic imaging methods used for hydrocarbon exploration. In the field of seismic hazard mitigation, the elastic properties of soil and geologic formations at the surface of the earth are a critical element in assessing and mitigating hazards in areas prone to strong earthquake shaking. In all these fields, it is useful and often essential to have an efficient, reliable means to test elastic properties of the materials in question.
It is well known that the elastic properties of a material can be tested by measuring the elastic response of a material to an applied dynamic driving force. In laboratory tests, a testing apparatus typically constrains a sample of the material to restrict the elastic response to certain defined modes or directions of motion, applies known confining pressures, and provides a means to measure the dynamic dimensions and shape of the body of material as needed. However, for many important types of materials it is time consuming, costly, or impractical to constrain the material in a test apparatus. Also, for some materials the process of isolating a sample of the material changes the elastic properties, a particular example being soils. There is therefore a need to efficiently and reliably measure the elastic properties of an unconstrained, unaltered material. Methods have previously been proposed for testing an unconstrained material by measuring the motion of the material at the same point where a measured dynamic driving force is concurrently applied. Because the motion measurement and force application are at the same place and time, the driving-point measurement process is very time efficient and access to only one measurement point is needed. However, it is a difficult problem to reliably determine elastic properties from the unconstrained driving-point motions of a real material.
For an ideal half-space of isotropic linear elastic material, it has previously been shown that the driving-point motion of the material comprises both transient and steady-state motions that are a complicated function of a combination of three independent elastic properties as well as being a function of frequency. Also, real materials are typically nonlinear and respond with anharmonic motions comprising a set of harmonic frequency components, and for highly nonlinear materials the anharmonic frequency components can dominate the response. Current practices for determining elastic properties from a measured driving-point response are typically variations of a LaPlace transform type analysis, wherein the complex ratio of the driving force to the measured motion is analyzed to determine a stiffness and/or viscosity of the material. The LaPlace type methods are based on a mathematical assumption of a sinusoidal driving force, assumption of a small-amplitude approximation for the motion of a linear material, and an assumption of a steady-state response. These assumptions limit the usefulness of these methods to low-amplitude driving forces applied for sufficiently long duration to approximate a steady-state response. Most of these assumptions are violated to some degree in real measurements. Because the low-amplitude measurements are more sensitive to measurement error and noise contamination, another current practice is determine the elastic properties based on an average value over a wide range of frequencies. Another practice known in the art is to attenuate both noise and anharmonic frequency components using a tracking filter. The low-amplitude limitation also restricts the usefulness in fields where larger driving-forces might otherwise be advantageous. For example, seismic vibrators used in seismic surveying produce very large driving forces far in excess of the low-amplitude limitations of the existing analysis methods. Hamblen et al. (U.S. Pat. No. 6,604,432) provide a method using a LaPlace-type analysis for estimating soil stiffness from a measured driving-point response, wherein the driving force must be limited to a low level, and the stiffness is averaged over a wide range of frequencies. The problem is that the stiffness and viscosity of a real, unconstrained material are a complicated function of frequency, and may not be well represented by an average value over a wide range of frequencies. Furthermore, the transient and anharmonic response components represent useful information about the elastic properties of real, nonlinear elastic material, so attenuating these components causes a loss of useful information. The transient components and anharmonic frequency components can represent a substantial portion of the total elastic energy in the measured motion of the driving point motion, and attenuating these components results in a misrepresentation of the relationship of the driving-force to the motion. Therefore, existing practices for analyzing a measured driving-point response to determine elastic properties of a material are of questionable reliability and subject to a number of substantial limitations.
The present invention provides systems and methods to use a measured driving-point response of a nonlinear material to determine one or more elastic properties of the material. The present invention takes advantage of the full information represented by the transient component, the steady-state component, the anharmonic components, and the nonlinear response components of a measured driving-point response of a real nonlinear material, without limitation in the use of large-amplitude forces. The elastic properties are determined by forming and solving a time-domain system of linear equations representing a differential equation model of the driving-point motions of the material. The time-domain differential equation model is free of the mathematical assumptions and limitations of existing methods. Based on a single, short duration, large-amplitude driving point measurement, both linear and nonlinear properties can be determined; both large-amplitude and near-zero amplitude properties can be determined; and elastic-wave speed and elastic moduli and their variation with depth can be determined.